The CCR estimator discussed here is a straightforward extension of Park and Hahn (1999) to accommodate an additional I(0) regressor (st). In order to avoid excessive notation in
our exposition on estimation and testing procedures for the TVC regression in (4), we make a simplifying assumption that τ = 0. We also assume that st∼ I(0) and uncorrelated with
the error and that a cointegrating vector of (yt, xt, pt)′ exists.
Defining wt= (ut, △xt, △pt)′= (w1t, w2t′ ) ′
, we assume that an invariance principle holds for wt, such that T−1/2P[T r]t=1wt →d B(r), where B = BM (Ω) is Brownian motion with
variance Ω. As is well known, Ω = P∞
k=−∞Ewtwt−k′ , and we define Σ = Ewtwt′ and
∆ = P∞
k=0Ewtwt−k′ , as is typical in this literature. We employ a typical partition of the
long-run variance given by
Ω = " ω11 ω12 ω21 Ω22 # ,
etc., and ∆ and Σ are partitioned accordingly.
Estimating a feasible CCR is a two-step procedure. We first estimate the model in (4) using least squares and construct consistent estimates of the variances above. Standard consistent long-run variance estimators are acceptable, and we use a nonparametric esti- mator with Parzen window and Andrews lag truncation. We use ˜θ, ˜Σ, etc. to denote the consistent estimates from this step.
We then construct z∗ pqt≡ φpq(t/T ) 0 0 0 1 0 0 0 1 xt pt st − " (˜δ′ 12, ˜∆ ′ 22) ˜Σ −1 wt 0 # ,
a (p + 2q + 3) × 1 canonical regressor vector, and
y∗ t ≡ yt− w′tΣ˜ −1 (˜δ′ 12, ˜∆ ′ 22) ′ φpq(t/T )′ 0 0 0 1 0 0 0 1 ˜ θ − ˜ω12Ω˜−221w2t,
the canonical regressand. The canonical cointegrating regression, y∗ t = τ + z ∗′ pqtθ + u ∗ pqt, (13)
thus has an error term of u∗
pqt ≡ upqt − ˜ω12Ω˜−221w2t. This CCR may be estimated us-
ing least squares, but with standard errors estimated from a variance estimator given by ˜ ω∗2 P tz ∗ pqtzpqt∗′ −1
, where ˜ω2∗ is a consistent estimator of ω 2
∗ = ω11− ω12Ω −1
22ω21 using the
variance estimators from the first step regression.
The cointegration tests discussed above are variable addition tests based on that of Park (1990). Specifically, these are Wald tests given by
WT = ˜ω∗−2 X t(ˆu ∗ pqt)2− X t(ˆu ∗s pqt)2 , where (ˆu∗
pqt) is a series of fitted residuals from the CCR regression in (13) and (ˆu∗pqts ) is
a series of fitted residuals from a CCR regression based on augmenting that in (13) with additional variables that have non-zero coefficients only under the alternative hypothesis of no cointegration. We use trend polynomials suggested by Park (1990). The limiting distribution is χ2 with degrees of freedom given by the number of variables added.
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